Do I have the choose to live with my dad in New York even though my mother lives in Ohio and has full custody?

No, Ohio does not let the child choose. It's a best interest standard. Your opinion goes into the factors but is not the final determinative factor.

When to file for custody?

File now. A parent trumps a grandparent in custody unless you are an unfit parent.

Does my sons name have to reflect his father being put on the birth certificate?

The only requirement is that the fathers name is put on the certificate as the father. The child's last name does not change unless you have agreed...

Is it legal to withhold a child from his father if I see him as unfit?

If there are no court orders for temporary visitation you would not legally be violating a court order. Be careful though your definition of unfit...

Can I file for abandonedment on the birth certificats father, he didn't show up for 3 paternity test.

If he's on the birth certificate why do you need a paternity test? Just file for child support.

How can my daughters lawyer tell her the father of her child will automatically get standard visition at the pretrial.

Given the facts you have stated are true, he's not likely to get any visitation unless it's supervised. She should ask her lawyer why they feel dad...

What city/ county would I file for my daughters name change in?

The county that you live in currently.

Are you not fit to be a parent? If yes, then yes she can if you are a fit parent then no she won't get custody parents trump next of kin.

You simply file a complaint for divorce in the common pleas court in your county and have him served. Forms are available on the Ohio supreme...

What rights does an unmarried father have to custody of their child if the mother will not allow the father to see the child?

The only way he can get any rights is to file for them in the county where mom and the child live.